Finding Numbers: Equation For Sum Of Number & Square

Hey math enthusiasts! Let's dive into a classic problem: finding a number where the sum of that number and its square equals 42. This is a great example of how algebra can help us solve real-world (or at least, puzzle-world) scenarios. The question is, which equation helps us crack this code? Don't worry, we'll break it down step by step, so even if algebra isn't your favorite, you'll be nodding along by the end.

Understanding the Problem: Decoding the Words

First off, let's translate the problem into mathematical terms. When we see "the sum of a number," we can represent that unknown number with a variable. Let's use x. "Its square" means x multiplied by itself, or x². And "is 42" translates to the equals sign (=) followed by 42. So, what the problem is really saying is: x + x² = 42. Pretty straightforward, right?

Now, the tricky part is spotting which of the multiple-choice options matches this equation. Sometimes, the way the equation is written can be a little different, but it's all about understanding the core relationship. The key here is to recognize the components: a number (x), its square (x²), and the sum of the two equaling 42. The equation is a way of stating that specific relationship, and from that equation, we will find the numbers!

This kind of problem is fundamental to algebra, and being able to quickly interpret the words into a usable equation will help you a lot in all your future math endeavors. It is essential to develop this skill. Let's make sure we find the numbers to match the equation!

Breaking Down the Answer Choices: Finding the Right Fit

Okay, let's dissect the answer choices. Remember our goal: we're looking for an equation that represents x + x² = 42.

• Option 1: $x^2 + x = 42$: Boom! This is it, guys. This equation directly reflects the problem statement. The order might be slightly different (x² + x instead of x + x²), but addition is commutative, meaning the order doesn't change the outcome. This option perfectly matches our initial translation.
• Option 2: $x^2 + 2x = 42$: This one introduces a "2x". This would imply that instead of adding the number and its square, we are adding twice the number to its square. This isn't what the original problem is asking for, so we can discard this answer.
• Option 3: $x^2 + x + 42 = 0$: This one is close but not quite. It's got the x² and the x, but it also includes the 42, which is moved to the left side of the equation, making it equal to zero. If we rearranged it, we'd have x² + x = -42. That is not the same as our original equation.
• Option 4: $x^2 + 2x + 42 = 0$: Similar to Option 2, this equation isn't what we are looking for. It adds a 2x to the square, which isn't the original problem. Also, this one is made equal to zero, which means we will find a different number.

So, by carefully comparing the equations with our understanding of the problem, we can confidently pick the correct answer. The critical thing here is recognizing the relationship that the equation represents.

Solving the Equation: Finding the Numbers

Now that we've found the correct equation, let's quickly touch on how you'd solve it to find the actual numbers. The equation is x² + x = 42. To solve it, we need to rearrange it into a standard quadratic form (ax² + bx + c = 0). This is done by subtracting 42 from both sides:

x² + x - 42 = 0

From here, we can either factor the quadratic equation or use the quadratic formula to find the values of x. Factoring involves finding two numbers that multiply to -42 and add up to 1 (the coefficient of x). The numbers are 7 and -6. So, we can factor the equation as:

(x + 7)(x - 6) = 0

This gives us two possible solutions: x = -7 and x = 6. Let's check our answers to make sure.

For x = -7: (-7)² + (-7) = 49 - 7 = 42. That checks out!

For x = 6: 6² + 6 = 36 + 6 = 42. Yep, that one works too!

So, the two numbers that satisfy the original problem are -7 and 6. This is the fun part, where you see the equation come to life! You may be able to solve the equation very easily; if not, just try and understand the methods so that you can apply them at a later time.

Why This Matters: The Power of Algebra

This kind of problem might seem like a simple exercise, but it demonstrates the power of algebra. Being able to translate a word problem into an equation and solve it is a crucial skill in all areas of mathematics and beyond. This approach is used in physics, engineering, computer science, and even in fields like economics. The ability to identify patterns, represent them mathematically, and then solve for unknown values is a fundamental skill.

This seemingly simple equation is a starting point for so many more complex problems. Whether you're interested in the sciences, finance, or just like a good puzzle, understanding how to set up and solve algebraic equations will be useful. Keep practicing, and the skills will come with time!

Tips for Success: Mastering Word Problems

• Read Carefully: The most important thing is to read the problem at least twice. Make sure you understand what's being asked. Highlight or underline important information.
• Define Variables: Assign variables to the unknowns. For example, use x for the number we're trying to find. Make sure you know what each variable represents.
• Translate into an Equation: Break the problem down piece by piece. Translate each phrase into a mathematical expression or equation.
• Solve the Equation: Use the appropriate methods to solve the equation. This might involve factoring, using the quadratic formula, or other techniques.
• Check Your Answer: Always plug your answer back into the original problem to make sure it makes sense.

Conclusion: You've Got This!

So, there you have it, guys. We've tackled the problem, found the correct equation, and even solved for the numbers. Remember, practice is key. The more you work with these types of problems, the easier they will become. Don't be afraid to ask for help, and always remember to break down the problem into smaller, manageable steps. Keep practicing your skills, and soon you'll be solving these problems in your head! You got this! Keep on learning and expanding your math skills! Remember that even if it seems hard at first, it will become easier with effort and dedication.